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42 Cards in this Set
 Front
 Back
random phenomenon

individual outcomes are uncertain but there is a regular distribution of outcomes in a large number of repetitions


probability

proportion of times the outcome would occur in a long series of repetitions


independent trials

the outcome of one trial must not influence the outcome of any other


Sample space S (random phenomenon)

The set of all possible outcomes


Event

an outcome or a set of outcomes of a random phenomenon (a subset of the sample space)


4 Probability rules

1. 0 ≤ P(A) ≤ 1
2. P(S) = 1 3. Disjoint events (no outcomes in common)P(A or B) = P(A) + P(B) 4. Complement rule P(Ac) = 1  P(A) 

Disjoint events

Events that have no outcomes in common


Probability of any event (description)

Equals the sum of the probabilities of the outcomes making up the event


Probability of any event A (equation)

P(A) = (count of outcomes in A) / (count of outcomes in S)


Multiplication rule for independent events

Two events A and B are independent if knowing that one occurs does not change the probability that the other occurs so P(A and B) = P(A)P(B)


Disjoint and independet

Disjoint events cannot be independent (they can't occur together, so the fact that A occurs already says that B cannot occur)


Random variable

A variable whose value is a numerical outcome of a random phenomenon


Discrete random variable (X)

A random variable with a finite number of possible values


Probability distribution of discrete X

Lists the values and their probabilities


Probabilities of values of X (must satisfy these two things)

1. Every probability pi is a number between 0 and 1
2. p1 + p2 + ... + pk = 1 (probability of any event comprising any of the xi values is found by adding up their respective pi values) 

Probability histograms

Compare the probability of certain numerical outcomes


Continuous random variable X

Takes all values in an interval of numbers


Probability distribution of continuous X

Described by a density curve; the probability of any event is the area under the density curve and above the values of X that make up the event


Continuous probability distributions and individual outcomes

All are assigned probabilities of zero


Continuous probaility of normal distribution

Calculate probabilities using zscores
Z = (X  µ) / σ 

Mean of discrete random variable X

Σxipi = µx = x1p1 + x2p2 + ... + xnpn


Law of large numbers

Draw independent observations at random from any population with finite mean µ. Decide how accurately you would like to estimate µ. As the number of observations drawn increases, the mean xbar of the observed values eventually approaches the mean µ of the population as closely as you specified and then stays that close.


Variability of the random outcomes

The more variable the outcomes, the more trials are needed to ensure that mean outcome xbar close to the distribution mean µ


Linear transformation of the mean of a random variable

µ(new) = a + bµ


Addition of random variable means

µ(x+y) = µ(x) +µ(y)


Variance of a discrete random variable

Σ(xi  µx)^2(pi)
(standard dev, is sqrt) 

Independence and adding variance

Variances only add when there is no association between variables


Variance of the sum of two nonindependent variables

Depends on the correlation between the variables (it is nonzero whenever two random variables are not independent)


Linear transformation (a, b) for variances

σ^2 (new) = b^2σ^2


σ^2(X+Y) or σ^2(XY), independent equals

σ^2(X) +σ^2(Y)


Adding variances with correlation p

(X+Y)>add 2pσ(X)σ(Y)
(XY)>sub. 2pσ(X)σ(Y) 

Union (of collection of events

The event that at least one of the collection occurs


P(one or more of A, B, C)

P(A) + P(B) + P(C)


P(A or B) equals

P(A) + P(B)  P(A and B)


Conditional probability

Computing probability given the fact that some event has already happened


Multiplication rule: P(A and B) equals

P(A)P(B l A)
(P(B) given A) 

Conditional probability (P(A)>0)

P(B l A) = P(A and B) / P(A)


Intersection

Event that all of the events occur (out of any collection of events)


P(A and B and C)

P(A)P(B l A)P(C l A and B)


Tree diagram

(A and B)>multiply each nodal probability along a branch
P(B l A)>calculate probability starting from the node where A is decided 

Independent events (conditional probability)

P(B l A) = P(B)


Baye's Rule

A1, A2, ..., Ak are disjoint events without P=0 and add to 1
P(Ai l C) = [P(C l Ai)P(Ai)] / [P(C l A1)P(A1) + ... + P(Ak)P(C l Ak) C is another event whose probability is not 0 or 1 